Revista Brasileira de Meteorologia, v.23, n.1, 35-40, 2008
DOWNWARD SURFACE FLUX COMPUTATIONS IN A VERTICALLY INHOMOGENEOUS
GREY PLANETARY ATMOSPHERE
Marcos Pimenta de Abreu
Departamento de Modelagem Computacional,
Instituto Politécnico, Universidade do Estado do Rio de Janeiro;
Rua Alberto Rangel s/n; 28601-970 Nova Friburgo RJ; Brasil
E-mail: [email protected]
Received May 2006 - Accept March 2007
ABSTRACT
We describe an efficient computational scheme for downward surface flux computations in a
vertically inhomogeneous grey planetary atmosphere for different values of solar zenith angle. We
start with the basic equations of a recently developed discrete ordinates spectral nodal method, and
we derive suitable bidirectional functions whose diffuse components do not depend on the solar
zenith angle. We then make use of these bidirectional functions to construct an efficient scheme for
computing the downward surface fluxes in a given model atmosphere for a number of solar zenith
angles. We illustrate the merit of the computational scheme described here with downward surface
flux computations in a three-layer grey model atmosphere for four values of solar zenith angle, and
we conclude this article with general remarks and directions for future work.
Keywords: atmospheric radiation, discrete ordinates, spectral nodal method, bidirectional functions,
radiative flux, computational efficiency.
RESUMO: CÁLCULO DO FLUXO RADIATIVO SUPERFICIAL EM UMA ATMOSFERA
PLANETÁRIA CINZA E VERTICALMENTE NÃO-HOMOGÊNEA
Este artigo descreve um esquema computacional baseado em desenvolvimentos recentes do método
espectro-nodal de ordenadas discretas para o cálculo eficiente do fluxo radiativo superficial em uma
atmosfera planetária cinza e verticalmente não-homogênea para valores distintos do ângulo zenital
solar. A partir das equações básicas do método espectro-nodal de ordenadas discretas, são obtidas
funções bidirecionais discretas cujas componentes difusas não dependem do ângulo zenital solar.
Com essas funções bidirecionais discretas, é construído um esquema computacional para calcular
eficientemente fluxos radiativos superficiais em uma dada atmosfera-modelo para vários ângulos
zenitais solares. O mérito computacional do esquema resultante é ilustrado com resultados numéricos
para os fluxos radiativos superficiais em uma atmosfera-modelo cinza com três camadas para quatro
valores distintos do ângulo zenital solar. Este artigo é finalizado com observações gerais e indicações
de trabalhos futuros.
Palavras-chave: radiação atmosférica, ordenadas discretas, método espectro-nodal, funções
bidirecionais, fluxo radiativo, eficiência computacional.
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1. INTRODUCTION
2. THE GOVERNING EQUATIONS
We have recently developed a discrete ordinates (DOR)
methodology for efficiently and accurately solving radiative
transfer problems on a stratified plane-parallel (multislab)
grey planetary atmosphere (DE ABREU, 2004a,b, 2005a). In
a continuous effort to further develop our methodology, we
have recently derived discrete ordinates bidirectional functions
for the boundary layers of a multislab model atmosphere (DE
ABREU, 2005b,c). These bidirectional functions are response
matrix-like relations that can be used to replace the boundary
layers in multislab radiative transfer computations without
degrading the numerical results. Moreover, they have shown to
increase the efficiency of our DOR methodology in solving a
set of multislab problems in which the optical properties of the
boundary layers do not change from one problem to another in
the set.
More recently, we have derived discrete ordinates
bidirectional functions for an internal layer (DE ABREU,
2006). As with our boundary layer functions, our internal layer
functions are response matrix-like relations that can be used to
replace without loss of accuracy an internal layer in multislab
radiative transfer computations, and they have also shown to
increase the efficiency of our DOR methodology in solving a
set of multislab problems with an optically stationary internal
layer.
In this article, we briefly discuss our discrete
bidirectional functions and, in connection to our DOR
methodology, we describe a computational scheme for
efficiently computing discrete ordinates approximations to
the downward surface fluxes (CHANDRASEKHAR, 1950)
in a vertically inhomogeneous grey model atmosphere for a
number of solar zenith angles. Efficient and accurate downward
surface flux computations are central to a number of relevant
solar-atmospheric applications such as environmental remote
sensing (THOMAS AND STAMNES, 1999; LIOU, 2002),
solar radiation dosimetry (KRZYSCIN ET AL., 2003;
BOLDEMANN ET AL., 2006), and solar power plant design
(SEGAL AND EPSTEIN, 2003; FRICKER, 2004).
An outline of the remainder of this article follows.
In Section 2, we introduce the governing equations of the
atmospheric radiation problems considered in this article. In
Section 3, we briefly discuss our discrete bidirectional functions
with the help of adequate referencing, and we show how they can
be used to construct an efficient scheme for downward surface
flux computations in a grey model atmosphere for a number of
solar zenith angles. In Section 4, we present numerical results
for downward surface fluxes for a three-layer model atmosphere,
and in Section 5 we close this article with concluding remarks
and directions for further research.
Let us consider a vertically inhomogeneous grey
planetary atmosphere illuminated with a monodirectional beam
of solar radiation (CHANDRASEKHAR, 1950). We assume
that the atmosphere overlies a dark surface of a terrestrial
planet, and that it can be represented by a multislab domain
consisting of R contiguous, disjoint, and uniform layers each
so that layer 1 (one) is the uppermost layer and layer R is the
lowermost one. We assume further that the frequency-integrated
diffuse component of the radiation field in the shortwave can
be accurately described by the frequency-integrated discrete
ordinates scalar equations (DE ABREU, 2004a)
(1)
where the integer N is the order of the angular quadrature,
m =1 : N/2 holds for the downward directions µm > 0, m =
N/2 +1 : N holds for the upward directions µm < 0, and the
remaining notation in equations (1) is standard. Details about
the uncollided component of the radiation field can be found
elsewhere (THOMAS AND STAMNES, 1999; LIOU, 2002).
We assume that changes in the refractive index are negligible
between two adjacent layers, and so continuity conditions
for the diffuse intensity at layer interfaces can be considered.
In addition, we use the discrete homogeneous boundary
conditions
(2)
Equations (1) and (2) can be numerically solved with
our new DOR methodology without optical truncation error.
A detailed discussion on the advantages and limitations of
this methodology can be found in earlier work (DE ABREU,
2004a,b, 2005a). One of the advantages is that it provides us
with the means of deriving nonstandard discrete bidirectional
functions, which replace without loss of accuracy an atmospheric
layer in radiative transfer computations, as we shall see in the
next section.
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Revista Brasileira de Meteorologia
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3. DISCRETE BIDIRECTIONAL FUNCTIONS
AND A COMPUTATIONAL SCHEME
We begin this section by setting forth the equations of
our DOR methodology. These equations are two sets of optically
discretized equations. The first set consists of the radiative heat
balance equations
(7)
A detailed description of the numerical schemes employed for
(3)
where Dtr L tr -tr–1 is the optical thickness of layer r; and
Idr-1,m Idr,m are the diffuse intensities in direction m at top and
bottom of layer r, respectively,
determining the coefficients qr,m,p and g 0r ,m in equations (6)
and (7) can be found in earlier work (DE ABREU, 2004a,b,
2005a).
Equations (3) and (6) are the starting point for the
derivation of our discrete bidirectional functions. Upon
substitution of equations (6) into equations (3) for an arbitrary
layer r, and after half-range splitting and suitable reformulation,
we get the discrete bidirectional functions:
(4)
is the rth layer-average diffuse intensity in direction m, and:
(8)
and
(5)
is the rth layer-average first collided source for direction m.
The quantity (tr–1–t0) in expression (5) is the relative optical
depth of layer r. The second set consists of the spectral auxiliary
equations
(6)
where
(9)
0
where definitions for Hr,m,p and Q r ,m can be found in DE
ABREU (2006). The discrete bidirectional functions (8) and
(9) give us the emerging diffuse intensities at the bottom and
top edges of layer r, respectively, in terms of the incident
diffuse intensities at top and bottom and of corresponding first
collided source terms. For r varying from two to R–1, these
bidirectional functions play for each r the role of discontinuous
conditions, and can advantageously be used to replace an
internal layer in some multislab radiative transfer computations
(DE ABREU, 2006). For r = R and in view of the discrete
boundary conditions (2) at tr, the second summation terms in the
bidirectional functions (8) and (9) vanish, and so we may write:
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(10)
and
(11)
The coefficients HR,m,p in the discrete bidirectional
functions (10) and (11) play the role of discrete bidirectional
transmittance and reflectance functions, respectively, for the
lowermost layer of a multislab model atmosphere. As in the case
of an internal layer, the discrete bidirectional functions (10) and
(11) can advantageously be used to replace the lowermost layer
in some multislab radiative transfer computations (DE ABREU,
2005b). For r = 1 and in view of the discrete boundary conditions
(2) at t0, the first summation terms in the bidirectional functions
(8) and (9) vanish, and so we may write:
following: we compute and store the coefficients Hr,m,p for all
layers of the multislab model atmosphere. We then replace
the layers with the bidirectional functions (8–9), (10–11), or
(12–13), where appropriate, and we compute the downward
surface flux with our DOR methodology, as described in DE
ABREU (2004a), for each solar zenith angle. We do not have
to recompute the coefficients Hr,m,p each time the downward
surface flux is computed with our DOR methodology for a new
value of solar zenith angle. This integrated strategy prevents
unnecessary (re)computation of available quantities (Hr,m,p),
and thereby increases the efficiency of our DOR methodology
for downward surface flux computations, as we shall illustrate
next.
4. NUMERICAL RESULTS
To illustrate the computational scheme outlined in
the previous section for efficient downward surface flux
computations, we consider a three-layer model (R = 3) for a
vertically inhomogeneous grey atmosphere: a conservative
tropospheric air layer (r = 2, Dt2 = 1, v2 = 1) is surrounded by
an overlying background stratospheric aerosol layer
(r = 1, Dt1 = 0.2, v1 = 0.95) and by an underlying land haze
layer (r = 3, Dt3 = 0.3, v3 = 0.9). A greytone representation of
this grey model atmosphere is shown in Figure 1.
(12)
and
(13)
The coefficients H1,m,p in the discrete bidirectional
functions (12) and (13) play the role of discrete bidirectional
reflectance and transmittance functions, respectively, for the
uppermost layer of a multislab model atmosphere. As in the
lowermost layer case, the discrete bidirectional functions (12)
and (13) can advantageously be used to replace the uppermost
layer in some multislab radiative transfer computations (DE
ABREU, 2005c).
Let us now consider an application basic to radiative
transfer in planetary atmospheres. Suppose that we wish to
compute a discrete ordinates approximation to the downward
surface flux in a vertically inhomogeneous grey atmosphere for
a number of solar zenith angles. Since the coefficients Hr,m,p
depend only on the optical properties of the corresponding
layers and on the quadrature points and weights used in the
discrete ordinates scalar equations (1) and (2), we can do the
Figure 1 - A three-layer model atmosphere
We assume that the aerosol layer consists of spherical
(Lorentz-Mie) particles with an asymmetry factor of 0.9 (LIOU,
2002). The Legendre scattering phase function data for the
tropospheric layer (L2 = 8) and for the haze layer (L3 = 82) were
extracted from a work of GARCIA AND SIEWERT (1985). The
Legendre scattering data for the stratospheric aerosol layer (we
set L1 = 100 for simplicity) can be computed from the Legendre
expansion of the Henyey-Greenstein phase function model for
the aerosol particles (LIOU, 2002).
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Revista Brasileira de Meteorologia
In Table 1, we present N = 200 results for the total
(uncollided plus diffuse) downward surface flux (F+) for four
different values of m0, the cosine of the solar zenith angle. In
Table 2, we show computer memory location and execution
time for the four runs with (integrated computation) and
without (independent computation) the scheme outlined in the
last paragraph of Section 3. We should note that the numerical
results presented in this section were obtained executing our
DOR computer program written in standard FORTRAN on an
IBM-compatible PC (1.4 GHz-clock Intel Pentium 4 processor,
256 Mbytes of RAM) running on a GNU/Linux platform.
In order to ensure the quality of our results, we note that
the numerical values in Table 1 all agree to within ±1 in the last
figures given with corresponding results generated on a very
fine-mesh grid by a (less efficient) discrete ordinates computer
program based on an unconditionally stable discretization
method – diamond difference – that is (still) widely used in
radiation transport computations (LEWIS AND MILLER,
1993). It is apparent that our computational scheme increased
the efficiency of our DOR methodology for downward surface
flux computations, since significant reductions in both computer
memory (a 31.6% reduction) and execution time (a 22.1%
reduction) were achieved.
Table 1. Downward surface flux (Wm-2)a.
Table 2. Computer memory and execution (CPU) time.
5. CONCLUDING REMARKS
We have described in connection to our DOR
methodology an efficient scheme for computing the downward
surface flux in a vertically inhomogeneous grey planetary
atmosphere for different values of solar zenith angle. We have
assumed that the planetary surface is dark, that the atmosphere
can be represented by a multi-layered medium, and that the
optical properties of the layers are known quantities. Then, we
used the basic equations of our DOR methodology to derive
discrete bidirectional functions for each layer. These functions
play the role of response matrices to incident diffuse radiation
and to first-collided sources. Since their diffuse components
(Hr,m,p) do not depend on the solar zenith angle, we were able
to use these bidirectional functions to construct an efficient
scheme for computing the downward surface fluxes for an
39
arbitrary number of solar zenith angles. We plan to extend this
computational scheme to the more realistic case of a reflecting
(rather than a dark) planetary surface, and we intend to report
our findings on these matters in a forthcoming article.
We conclude by noting that the discrete bidirectional
functions (8–13) can be incorporated into discrete ordinates
methodologies other than our DOR methodology, e.g. the
methods described by STAMNES ET AL. (1988), SIEWERT
(2000), and BARICHELLO AND SIEWERT (2001) as follows:
the discrete bidirectional functions (10–11) and/or (12–13), where
appropriate, can be used to replace corresponding boundary
layers at bottom and/or top, while the discrete bidirectional
functions (8–9) can be used to replace corresponding internal
layers, playing the role of fictitious discontinuous conditions
(DE ABREU, 2006) for the diffuse intensity of the radiation
field.
6. ACKNOWLEDGMENTS
The author is gratefully acknowledged to Prof. Paul
Hardaker (Met Office, UK) for constructive criticisms regarding
this (and related) work, and to three anonymous referees for
many valuable suggestions.
7. REFERENCES
BARICHELLO, L. B.; SIEWERT, C. E. A new version of
the discrete-ordinates method. In: INTERNATIONAL
CONFERENCE ON COMPUTATIONAL HEAT AND
MASS TRANSFER, 2., October 22–26, 2001. COPPE/
EE/UFRJ, Federal University of Rio de Janeiro, Brazil.
Proceedings of the 2nd International Conference on
Computational Heat and Mass Transfer, Rio de Janeiro: Epapers Serviços Editoriais Ltda., 2001. 1 CD.
BOLDEMANN, C.; BLENNOW, M.; DAL, H.; MARTENSSON,
F.; RAUSTORP, A.; YUEN, K.; WESTER, U. Impact of
preschool environment upon children’s physical activity and
sun exposure. Prev. Med., v. 42, n. 4, p. 301–308, 2006.
CHANDRASEKHAR, S. Radiative Transfer. Oxford: Clarendon
Press, 1950. 393 p.
DE ABREU, M. P. A two-component method for solving
multislab problems in radiative transfer. J. Quant. Spectrosc.
Radiat. Transfer, v. 85, n. 3/4, p. 311–336, 2004a.
DE ABREU, M. P. Mixed singular-regular boundary conditions
in multislab radiation transport. J. Comp. Phys., v. 197, n.
1, p. 167–185, 2004b.
DE ABREU, M. P. A mathematical method for solving mixed
problems in multislab radiative transfer. J. Braz. Soc. Mech.
Sci. Eng., v. 27, n. 4, p. 381–393, 2005a.
DE ABREU, M. P. Layer-edge conditions for multislab
40������������������������������������
Marcos Pimenta de Abreu�������������
Volume 23(1)
atmospheric radiative transfer problems. J. Quant. Spectrosc.
Radiat. Transfer, v. 94, n. 2, p. 137–149, 2005b.
DE ABREU, M. P. On efficiently solving a class of atmospheric
radiative transfer problems using discrete ordinates models
for bidirectional functions: the uppermost layer. Atmos. Sci.
Lett., v. 6, n. 2, p. 84–89, 2005c.
DE ABREU, M. P. On equivalent diffuse conditions for an
internal layer in multislab atmospheric radiation. J. Quant.
Spectrosc. Radiat. Transfer, v. 102, n. 3, p. 519–529,
2006.
FRICKER, H. W. Regenerative thermal storage in atmospheric
air system solar power plants. Energy, v. 29, n. 5/6, p.
871–881, 2004.
GARCIA, R. D. M.; SIEWERT, C. E. Benchmark results in
radiative transfer. Transp. Th. Stat. Phys., v. 14, n. 4, p.
437–483, 1985.
KRZYSCIN, J. W.; JAROSLAWSKI, J.; SOBOLEWSKI, P. S.
Effects of clouds on the surface erythemal UV-B irradiance
at northern midlatitudes: estimation from the observations
taken at Belsk, Poland (1999-2001). J. Atmos. Solar-Terrest.
Phys., v. 65, p. 457-467, 2003.
LEWIS, E. E.; MILLER JR., W. F. Computational Methods
of Neutron Transport. Second edition. Lagrange Park, IL:
American Nuclear Society, 1993. 401 p.
LIOU, K. –N. An Introduction to Atmospheric Radiation.
Second edition. New York: Academic Press, 2002. 597 p.
SEGAL, A.; EPSTEIN, M. Solar ground reformer. Solar Energy,
v. 75, n. 6, p. 479–490, 2003.
SIEWERT, C. E. A concise and accurate solution to
Chandrasekhar’s basic problem in radiative transfer. J.
Quant. Spectrosc. Radiat. Transfer, v. 64, n. 2, p. 109–130,
2000.
STAMNES, K.; TSAY, S.; WISCOMBE, W.; JAYAWEERA, K.
Numerically stable algorithm for discrete-ordinate-method
radiative transfer in multiple scattering and emitting layered
media. Appl. Optics, v. 27, n. 12, p. 2502–2509, 1988.
THOMAS, G. E.; STAMNES, K. Radiative Transfer in the
Atmosphere and Ocean. New York: Cambridge University
Press, 1999. 517 p.